Slides based on the book:

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

Their main occupation was forming various clubs,
which at some point started threatening the very survival of the city.

⚠️ There could be as many as \(2^n\) distinct clubs!

(Well, \(2^n - 1\) if you would prefer to exclude the empty club.)

In order to limit the number of clubs, the city council decreed the following innocent-looking rules:

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

Their main occupation was forming various clubs,
which at some point started threatening the very survival of the city.

⚠️ There could be as many as \(2^n\) distinct clubs!

(Well, \(2^n - 1\) if you would prefer to exclude the empty club.)

In order to limit the number of clubs, the city council decreed the following innocent-looking rules:

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

Their main occupation was forming various clubs,
which at some point started threatening the very survival of the city.

⚠️ There could be as many as \(2^n\) distinct clubs!

(Well, \(2^n - 1\) if you would prefer to exclude the empty club.)

In order to limit the number of clubs, the city council decreed the following innocent-looking rules:

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

Their main occupation was forming various clubs,
which at some point started threatening the very survival of the city.

⚠️ There could be as many as \(2^n\) distinct clubs!

(Well, \(2^n - 1\) if you would prefer to exclude the empty club.)

In order to limit the number of clubs, the city council decreed the following innocent-looking rules:

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(n\) clubs.

Example: \(\{\{1\},\{2\}, \ldots, \{n\}\}\).

(The Singletons Clubs.)

Example: \(\{\{1,2,3\},\{1,2,4\}, \ldots, \{1,2,n\}\}\).

(Where 1 and 2 are popular.)

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(n\) clubs.

subsets of \([n]\)

vectors in \(n\)-dimensional space

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(n\) clubs.

sets that satisfy (1) & (2)

linearly independent vectors

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(n\) clubs.

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(n\) clubs.

\(\in \mathbb{F}^n_2\)

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(n\) clubs.

\(\{1,3,5,6,7\} \subseteq [10] \longrightarrow (1,0,1,0,1,1,1,0,0,0)\)

Another Example:

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(n\) clubs.

\(\{{\color{IndianRed}1},{\color{DodgerBlue}3},{\color{SeaGreen}5},{\color{Tomato}6},{\color{Purple}7}\} \subseteq [10] \longrightarrow ({\color{IndianRed}1},0,{\color{DodgerBlue}1},0,{\color{SeaGreen}1},{\color{Tomato}1},{\color{Purple}1},0,0,0)\)

Another Example:

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(n\) clubs.

Claim. If \(\{S_1, \ldots, S_t\}\) forms a valid set of clubs over \([n]\),
then \(\{v_1, \ldots v_t\}\) is a linearly independent collection of vectors in \(\mathbb{F}_2^n\), where \(v_i := f(S_i)\).

\(\alpha_1 v_1 + \cdots + \alpha_i v_i + \cdots + \alpha_j v_j + \cdots + \alpha_t v_t = 0\)

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(n\) clubs.

\(\alpha_1 (v_1 \cdot v_i) + \cdots + \alpha_i (v_i \cdot v_i) + \cdots + \alpha_j (v_j \cdot v_i) + \cdots + \alpha_t (v_t \cdot v_i) = 0\)

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(n\) clubs.

\(\alpha_1 (v_1 \cdot v_i) + \cdots + \alpha_i ({\color{IndianRed}v_i \cdot v_i}) + \cdots + \alpha_j (v_j \cdot v_i) + \cdots + \alpha_t (v_t \cdot v_i) = 0\)

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(n\) clubs.

\( ({\color{IndianRed}1},0,{\color{DodgerBlue}1},0,{\color{SeaGreen}1},{\color{Tomato}1},{\color{Purple}1},0,0,0)\)

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(n\) clubs.

\(\alpha_1 (v_1 \cdot v_i) + \cdots + \alpha_i ({\color{IndianRed}v_i \cdot v_i}) + \cdots + \alpha_j (v_j \cdot v_i) + \cdots + \alpha_t (v_t \cdot v_i) = 0\)

\({\color{IndianRed}|S_i| \mod 2}\)

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(n\) clubs.

\(\alpha_1 (v_1 \cdot v_i) + \cdots + \alpha_i ({\color{IndianRed}v_i \cdot v_i}) + \cdots + \alpha_j ({\color{DodgerBlue}v_j \cdot v_i}) + \cdots + \alpha_t (v_t \cdot v_i) = 0\)

\({\color{IndianRed}|S_i| \mod 2}\)

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(n\) clubs.

\(\{{\color{IndianRed}1},{\color{DodgerBlue}3},{\color{Tomato}5},{\color{Tomato}6},{\color{Tomato}7}\} \subseteq [10] \longrightarrow ({\color{IndianRed}1},0,{\color{DodgerBlue}1},0,{\color{Tomato}1},{\color{Tomato}1},{\color{Tomato}1},0,0,0)\)

\(\{{\color{IndianRed}1},{\color{DodgerBlue}3},{\color{SeaGreen}4},{\color{SeaGreen}8},{\color{SeaGreen}9}\} \subseteq [10] \longrightarrow ({\color{IndianRed}1},0,{\color{DodgerBlue}1},{\color{SeaGreen}1},0,0,0,{\color{SeaGreen}1},{\color{SeaGreen}1},0)\)

\(({\color{IndianRed}1},0,{\color{DodgerBlue}1},0,0,0,0,0,0,0)\)

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(n\) clubs.

\(\alpha_1 (v_1 \cdot v_i) + \cdots + \alpha_i ({\color{IndianRed}v_i \cdot v_i}) + \cdots + \alpha_j ({\color{DodgerBlue}v_j \cdot v_i}) + \cdots + \alpha_t (v_t \cdot v_i) = 0\)

\({\color{IndianRed}|S_i| \mod 2}\)

\({\color{DodgerBlue}|S_i \cap S_j| \mod 2}\)

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(n\) clubs.

\(\alpha_1 (v_1 \cdot v_i) + \cdots + \alpha_i ({\color{IndianRed}v_i \cdot v_i}) + \cdots + \alpha_j ({\color{DodgerBlue}v_j \cdot v_i}) + \cdots + \alpha_t (v_t \cdot v_i) = 0\)

\({\color{IndianRed}1}\)

\({\color{DodgerBlue}0}\)

#3. The Clubs of OddTown

There are \(n\) citizens living in Oddtown.

(1) Each club has to have an odd number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(n\) clubs.

\({\color{Silver}\alpha_i (v_1 \cdot v_i) + \cdots +} \alpha_i ({\color{IndianRed}v_i \cdot v_i}){\color{Silver} + \cdots + \alpha_j (v_j \cdot v_i) + \cdots + \alpha_t (v_t \cdot v_i)} = 0\)

\(\implies \alpha_i = 0\), \(\forall i \in [n]\)

#3. The Clubs of OddTown

What about Eventown?

#3. The Clubs of OddTown

There are \(n\) citizens living in Eventown.

(1) Each club has to have an even number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(2^{\lfloor \frac{n}{2} \rfloor}\) clubs.

Claim. If \(\{S_1, \ldots, S_t\}\) is a maximal and valid set of clubs over \([n]\),
then \(\{v_1, \ldots v_t\}\) is a totally isotropic subspace of dimension at most \(\lfloor \frac{n}{2} \rfloor\).

Note that \(v_i \cdot v_j = 0\) for all \(1 \leqslant i,j \leqslant t\)

Let \(X := \{v_1, \ldots, v_t\}\).

In other words, \(X \perp X\), implying that \(X \subseteq X^{\perp}\).

#3. The Clubs of OddTown

There are \(n\) citizens living in Eventown.

(1) Each club has to have an even number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(2^{\lfloor \frac{n}{2} \rfloor}\) clubs.

Claim. If \(\{S_1, \ldots, S_t\}\) is a maximal and valid set of clubs over \([n]\),
then \(\{v_1, \ldots v_t\}\) is a totally isotropic subspace of dimension at most \(\lfloor \frac{n}{2} \rfloor\).

Let \(X := \{v_1, \ldots, v_t\}\).

In other words, \(X \perp X\), implying that \(X \subseteq X^{\perp}\).

If \(v\) is in span\((X)\), then \(v \perp X\): therefore \(X\) is closed
(since \(\{S_1, \ldots, S_t\}\) is maximal).

#3. The Clubs of OddTown

There are \(n\) citizens living in Eventown.

(1) Each club has to have an even number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(2^{\lfloor \frac{n}{2} \rfloor}\) clubs.

Claim. If \(\{S_1, \ldots, S_t\}\) is a maximal and valid set of clubs over \([n]\),
then \(\{v_1, \ldots v_t\}\) is a totally isotropic subspace of dimension at most \(\lfloor \frac{n}{2} \rfloor\).

Let \(X := \{v_1, \ldots, v_t\}\).

In other words, \(X \perp X\), implying that \(X \subseteq X^{\perp}\).

Therefore, \(X\) is a subspace.

#3. The Clubs of OddTown

There are \(n\) citizens living in Eventown.

(1) Each club has to have an even number of members.

(2) Every two clubs must have an even number of members in common.

Under these rules, it is impossible to form more than \(2^{\lfloor \frac{n}{2} \rfloor}\) clubs.

Claim. If \(\{S_1, \ldots, S_t\}\) is a maximal and valid set of clubs over \([n]\),
then \(\{v_1, \ldots v_t\}\) is a totally isotropic subspace of dimension at most \(\lfloor \frac{n}{2} \rfloor\).

Let \(X := \{v_1, \ldots, v_t\}\).

In other words, \(X \perp X\), implying that \(X \subseteq X^{\perp}\).

\(\dim(X) + \dim(X^\perp) = n \implies \dim(X) \leqslant \lfloor \frac{n}{2} \rfloor\).

#4. Same-Size Intersections

Generalized Fisher inequality.

If \(C_1, C_2, \ldots, C_m\) are
distinct and nonempty subsets
of an \(n\)-element set

such that

all the intersections \(C_i \cap C_j, i \neq j\),
have the same size (say \(t\)),

then

\(m \leqslant n\).

#4. Same-Size Intersections

Given: \(C_1, C_2, \ldots, C_m \in 2^{[n]}\)

where \(|C_i \cap C_j| = t\) for all \(1\leqslant i < j \leqslant m\)

To Prove: \(m \leqslant n\).

Case 1. \(|C_i| = t\) for some \(i \in [m]\).

At most \(n-t\) other sets.

Total # of sets \(\leqslant n-t + 1 \leqslant n\).

#4. Same-Size Intersections

Given: \(C_1, C_2, \ldots, C_m \in 2^{[n]}\)

where \(|C_i \cap C_j| = t\) for all \(1\leqslant i < j \leqslant m\)

To Prove: \(m \leqslant n\).

Case 2. \(|C_i| > t\) for all \(i \in [m]\).

\(a_{i j}= \begin{cases}1 & \text { if } j \in C_i, \text { and } \\ 0 & \text { otherwise }\end{cases}\)

Let \(A\) be the \(m \times n\) matrix with entries:

#4. Same-Size Intersections

Given: \(C_1, C_2, \ldots, C_m \in 2^{[n]}\)

where \(|C_i \cap C_j| = t\) for all \(1\leqslant i < j \leqslant m\)

To Prove: \(m \leqslant n\).

Case 2. \(|C_i| > t\) for all \(i \in [m]\).

\(\{{\color{IndianRed}1,2,5}\},\{{\color{DodgerBlue}2,3}\},\{{\color{DarkSeaGreen}3,4,5}\}\)

\(\left(\begin{array}{ccccc}{\color{IndianRed}1} & {\color{IndianRed}1} & 0 & 0 & {\color{IndianRed}1} \\0 & {\color{DodgerBlue}1} & {\color{DodgerBlue}1} & 0 & 0 \\ 0 & 0 & {\color{DarkSeaGreen}1} & {\color{DarkSeaGreen}1} & {\color{DarkSeaGreen}1} \end{array}\right)\)

#4. Same-Size Intersections

Given: \(C_1, C_2, \ldots, C_m \in 2^{[n]}\)

where \(|C_i \cap C_j| = t\) for all \(1\leqslant i < j \leqslant m\)

To Prove: \(m \leqslant n\).

Case 2. \(|C_i| > t\) for all \(i \in [m]\).

\(B := AA^T\)

\(=\)

\(A\)

\(A^T\)

\(B\)

\(C_i\)

\(C_j\)

#4. Same-Size Intersections

\(=\)

\(A\)

\(A^T\)

\(B\)

\(C_i\)

\(C_j\)

\(t\)

#4. Same-Size Intersections

\(=\)

\(A\)

\(A^T\)

\(B\)

\(C_i\)

#4. Same-Size Intersections

\(=\)

\(A\)

\(A^T\)

\(B\)

\(C_i\)

\(d_i\)

#4. Same-Size Intersections

Given: \(C_1, C_2, \ldots, C_m \in 2^{[n]}\)

where \(|C_i \cap C_j| = t\) for all \(1\leqslant i < j \leqslant m\)

To Prove: \(m \leqslant n\).

Case 2. \(|C_i| > t\) for all \(i \in [m]\).

\(B := AA^T\)

\(B=\left(\begin{array}{ccccc}d_1 & t & t & \ldots & t \\t & d_2 & t & \ldots & t \\\vdots & \vdots & \vdots & \vdots & \vdots \\t & t & t & \ldots & d_m\end{array}\right)\)

Recall that \(A\) is a \(m \times n\) matrix, so: rank\((A) \leqslant n\)

#4. Same-Size Intersections

Given: \(C_1, C_2, \ldots, C_m \in 2^{[n]}\)

where \(|C_i \cap C_j| = t\) for all \(1\leqslant i < j \leqslant m\)

To Prove: \(m \leqslant n\).

Case 2. \(|C_i| > t\) for all \(i \in [m]\).

\(B := AA^T\)

\(B=\left(\begin{array}{ccccc}d_1 & t & t & \ldots & t \\t & d_2 & t & \ldots & t \\\vdots & \vdots & \vdots & \vdots & \vdots \\t & t & t & \ldots & d_m\end{array}\right)\)

\({\color{White}m = }\)rank\((B) \leqslant \) rank\((A) \leqslant n\)

\({\color{White}m = }\)rank\((XY) \leqslant \) rank\((X)\)

#4. Same-Size Intersections

Given: \(C_1, C_2, \ldots, C_m \in 2^{[n]}\)

where \(|C_i \cap C_j| = t\) for all \(1\leqslant i < j \leqslant m\)

To Prove: \(m \leqslant n\).

Case 2. \(|C_i| > t\) for all \(i \in [m]\).

\(B := AA^T\)

\(B=\left(\begin{array}{ccccc}d_1 & t & t & \ldots & t \\t & d_2 & t & \ldots & t \\\vdots & \vdots & \vdots & \vdots & \vdots \\t & t & t & \ldots & d_m\end{array}\right)\)

\(m = \) rank\((B) \leqslant \) rank\((A) \leqslant n\)

Given: \(C_1, C_2, \ldots, C_m \in 2^{[n]}\)

where \(|C_i \cap C_j| = t\) for all \(1\leqslant i < j \leqslant m\)

To Prove: \(m \leqslant n\).

Case 2. \(|C_i| > t\) for all \(i \in [m]\).

\(B := AA^T\)

\(B=\left(\begin{array}{ccccc}d_1 & t & t & \ldots & t \\t & d_2 & t & \ldots & t \\\vdots & \vdots & \vdots & \vdots & \vdots \\t & t & t & \ldots & d_m\end{array}\right)\)

\({\color{red}m = }\) rank\({\color{red}(B)} \leqslant \) rank\((A) \leqslant n\)

#4. Same-Size Intersections

Let \(T: V \rightarrow W\) be a linear transformation between two vector spaces where \(T\)'s domain \(V\) is finite dimensional. Then

\(\operatorname{rank}(T)+\operatorname{nullity}(T)=\operatorname{dim} V\)

where \(\operatorname{rank}(T)\) is the rank of \(T\) (the dimension of its image)

and

nullity \((T)\) is the nullity of \(T\) (the dimension of its kernel).

Let \(T: V \rightarrow W\) be a linear transformation between two vector spaces where \(T\)'s domain \(V\) is finite dimensional. Then

\(\operatorname{rank}(T)+ {\color{IndianRed}\operatorname{nullity}(T)}=\operatorname{dim} V\)

where \(\operatorname{rank}(T)\) is the rank of \(T\) (the dimension of its image)

and

nullity \((T)\) is the nullity of \(T\) (the dimension of its kernel).

\(= 0\)

#4. Same-Size Intersections

\(B=\left(\begin{array}{ccccc}d_1 & t & t & \ldots & t \\t & d_2 & t & \ldots & t \\\vdots & \vdots & \vdots & \vdots & \vdots \\t & t & t & \ldots & d_m\end{array}\right)\)

Suffices to show: \(\mathbf{x}^T B \mathbf{x}>0 \text { for all nonzero } \mathbf{x} \in \mathbb{R}^m \text {. }\)

because once we have the above,

if \(B\mathbf{x} = \mathbf{0}\), then \(\mathbf{x}^TB\mathbf{x} = \mathbf{x}^T\mathbf{0} = 0\), hence \(\mathbf{x} = 0\).

#4. Same-Size Intersections

\(B=\left(\begin{array}{ccccc}d_1 & t & t & \ldots & t \\t & d_2 & t & \ldots & t \\\vdots & \vdots & \vdots & \vdots & \vdots \\t & t & t & \ldots & d_m\end{array}\right)\)

Suffices to show: \(\mathbf{x}^T B \mathbf{x}>0 \text { for all nonzero } \mathbf{x} \in \mathbb{R}^m \text {. }\)

We can write \(B=t J_n+D\), where \(J_n\) is the all 1's matrix and
\(D\) is the diagonal matrix with \(d_1-t, d_2-t, \ldots, d_n-t\) on the diagonal.

#4. Same-Size Intersections

\(B=\left(\begin{array}{ccccc}d_1 & t & t & \ldots & t \\t & d_2 & t & \ldots & t \\\vdots & \vdots & \vdots & \vdots & \vdots \\t & t & t & \ldots & d_m\end{array}\right)\)

\(\left(\begin{array}{ccccc}d_1 & t & t & t \\t & d_2 & t & t \\t & t & d_3 & t \\t & t & t & d_4\end{array}\right) = \)

\(\left(\begin{array}{ccccc}0 & ~t~ & ~t~ & ~t~ \\t & ~0~ & ~t~ & ~t~ \\t & ~t~ & ~0~ & ~t~ \\t & ~t~ & ~t~ & ~0~ \end{array}\right)\)

+ \(\left(\begin{array}{ccccc}d_1 & 0 & 0 & 0 \\0 & d_2 & 0 & 0 \\0 & 0 & d_3 & 0 \\0 & 0 & 0 & d_4\end{array}\right) \)

#4. Same-Size Intersections

\(B=\left(\begin{array}{ccccc}d_1 & t & t & \ldots & t \\t & d_2 & t & \ldots & t \\\vdots & \vdots & \vdots & \vdots & \vdots \\t & t & t & \ldots & d_m\end{array}\right)\)

\(\left(\begin{array}{ccccc}~~~~~~t~~~~~~ & ~~~~~~t~~~~~~ & ~~~~~~t~~~~~~ & ~~~~~~t~~~~~~ \\~~~~~~t~~~~~~ & ~~~~~~t~~~~~~ & ~~~~~~t~~~~~~ & ~~~~~~t~~~~~~ \\~~~~~~t~~~~~~ & ~~~~~~t~~~~~~ & ~~~~~~t~~~~~~ & ~~~~~~t~~~~~~ \\~~~~~~t~~~~~~ & ~~~~~~t~~~~~~ & ~~~~~t~~~~~ & ~~~~~t~~~~~ \end{array}\right)\)

+ \(\left(\begin{array}{ccccc}(d_1-t) & 0 & 0 & 0 \\0 & (d_2-t) & 0 & 0 \\0 & 0 & (d_3-t) & 0 \\0 & 0 & 0 & (d_4-t)\end{array}\right) \)

\(\left(\begin{array}{ccccc}d_1 & t & t & t \\t & d_2 & t & t \\t & t & d_3 & t \\t & t & t & d_4\end{array}\right) = \)

#4. Same-Size Intersections

\(B=\left(\begin{array}{ccccc}d_1 & t & t & \ldots & t \\t & d_2 & t & \ldots & t \\\vdots & \vdots & \vdots & \vdots & \vdots \\t & t & t & \ldots & d_m\end{array}\right)\)

Suffices to show: \(\mathbf{x}^T B \mathbf{x}>0 \text { for all nonzero } \mathbf{x} \in \mathbb{R}^m \text {. }\)

We can write \(B=t J_n+D\), where \(J_n\) is the all 1's matrix and
\(D\) is the diagonal matrix with \(d_1-t, d_2-t, \ldots, d_n-t\) on the diagonal.

\(\mathbf{x}^T B \mathbf{x}=\mathbf{x}^T\left(t J_n+D\right) \mathbf{x}=t \mathbf{x}^T J_n \mathbf{x}+\mathbf{x}^T D \mathbf{x}\)

#4. Same-Size Intersections

\(B=\left(\begin{array}{ccccc}d_1 & t & t & \ldots & t \\t & d_2 & t & \ldots & t \\\vdots & \vdots & \vdots & \vdots & \vdots \\t & t & t & \ldots & d_m\end{array}\right)\)

Suffices to show: \(\mathbf{x}^T B \mathbf{x}>0 \text { for all nonzero } \mathbf{x} \in \mathbb{R}^m \text {. }\)

We can write \(B=t J_n+D\), where \(J_n\) is the all 1's matrix and
\(D\) is the diagonal matrix with \(d_1-t, d_2-t, \ldots, d_n-t\) on the diagonal.

\(\mathbf{x}^T B \mathbf{x}=\mathbf{x}^T\left(t J_n+D\right) \mathbf{x}=t {\color{IndianRed}\mathbf{x}^T J_n \mathbf{x}}+\mathbf{x}^T D \mathbf{x}\)

\({\color{IndianRed}\sum_{i, j=1}^n x_i x_j=\left(\sum_{i=1}^n x_i\right)^2 \geqslant 0}\)

#4. Same-Size Intersections

\(B=\left(\begin{array}{ccccc}d_1 & t & t & \ldots & t \\t & d_2 & t & \ldots & t \\\vdots & \vdots & \vdots & \vdots & \vdots \\t & t & t & \ldots & d_m\end{array}\right)\)

Suffices to show: \(\mathbf{x}^T B \mathbf{x}>0 \text { for all nonzero } \mathbf{x} \in \mathbb{R}^m \text {. }\)

We can write \(B=t J_n+D\), where \(J_n\) is the all 1's matrix and
\(D\) is the diagonal matrix with \(d_1-t, d_2-t, \ldots, d_n-t\) on the diagonal.

\(\mathbf{x}^T B \mathbf{x}=\mathbf{x}^T\left(t J_n+D\right) \mathbf{x}=t \mathbf{x}^T J_n \mathbf{x}+{\color{Olive}\mathbf{x}^T D \mathbf{x}}\)

\({\color{Olive}\mathbf{x}^T D \mathbf{x}=\sum_{i=1}^n\left(d_i-t\right) x_i^2>0}\)

#4. Same-Size Intersections

\(B=\left(\begin{array}{ccccc}d_1 & t & t & \ldots & t \\t & d_2 & t & \ldots & t \\\vdots & \vdots & \vdots & \vdots & \vdots \\t & t & t & \ldots & d_m\end{array}\right)\)

Suffices to show: \(\mathbf{x}^T B \mathbf{x}>0 \text { for all nonzero } \mathbf{x} \in \mathbb{R}^m \text {. }\)

We can write \(B=t J_n+D\), where \(J_n\) is the all 1's matrix and
\(D\) is the diagonal matrix with \(d_1-t, d_2-t, \ldots, d_n-t\) on the diagonal.

\(\mathbf{x}^T B \mathbf{x}=\mathbf{x}^T\left(t J_n+D\right) \mathbf{x}=t \mathbf{x}^T J_n \mathbf{x}+\mathbf{x}^T D \mathbf{x}\)

\(> 0\)

#4. Same-Size Intersections

Let \(L\) be a set of \(s\) nonnegative integers and
\(\mathscr{F}\) a family of subsets of an \(n\)-element set \(X\).

Suppose that for any two distinct members \(A, B \in \mathscr{F}\) we have \(|A \cap B| \in L\).

Assuming in addition that \(\mathscr{F}\) is uniform,
i.e. each member of \(\mathscr{F}\) has the same cardinality,
a celebrated theorem of D. K. Ray-Chaudhuri and R. M. Wilson asserts that:

\(|\mathscr{F}| \leqq\left(\begin{array}{l}n \\ s\end{array}\right)\).

#4. Same-Size Intersections

P. Frankl and R. M. Wilson proved that
without the uniformity assumption, we have:

\(|\mathscr{F}| \leqq\left(\begin{array}{l}n \\s\end{array}\right)+\left(\begin{array}{c}n \\s-1\end{array}\right)+\ldots+\left(\begin{array}{l}n \\0\end{array}\right)\)

#4. Same-Size Intersections

General Pattern of Question

17. Medium-Size Intersection Is Hard To Avoid

Suppose that \(\mathcal{F}\) is a system of subsets of an \(n\)-element set.

Suppose that certain simply described configuration of sets does not occur in \(\mathcal{F}\).

What is the maximum possible number of sets in \(\mathcal{F}\)?

The Sperner lemma: If there are no two distinct sets \(A, B \in \mathcal{F}\) with \(A \subset B\), then \(|\mathcal{F}| \leq\left(\begin{array}{c}n \\ \lfloor n / 2\rfloor\end{array}\right)\).

General Pattern of Question

17. Medium-Size Intersection Is Hard To Avoid

Suppose that \(\mathcal{F}\) is a system of subsets of an \(n\)-element set.

Suppose that certain simply described configuration of sets does not occur in \(\mathcal{F}\).

What is the maximum possible number of sets in \(\mathcal{F}\)?

The Erdős-Ko-Rado Theorem. If \(k \leqslant n / 2\), each \(A \in \mathcal{F}\) has exactly \(k\) elements, and \(A \cap B \neq \emptyset\) for every two \(A, B \in \mathcal{F}\), then \(|\mathcal{F}| \leq\left(\begin{array}{c}n-1 \\ k-1\end{array}\right)\).

General Pattern of Question

17. Medium-Size Intersection Is Hard To Avoid

Suppose that \(\mathcal{F}\) is a system of subsets of an \(n\)-element set.

Suppose that certain simply described configuration of sets does not occur in \(\mathcal{F}\).

What is the maximum possible number of sets in \(\mathcal{F}\)?

OddTown! EvenTown! etc. etc.

General Pattern of Question

17. Medium-Size Intersection Is Hard To Avoid

Suppose that \(\mathcal{F}\) is a system of subsets of an \(n\)-element set.

Suppose that certain simply described configuration of sets does not occur in \(\mathcal{F}\).

What is the maximum possible number of sets in \(\mathcal{F}\)?

Theorem. Let \(p\) be a prime number and let \(\mathcal{F}\) be a system
of \((2p-1)\) element subsets of an \(n\)-element set \(X\) such that

no two sets in \({\color{SeaGreen}\mathcal{F}}\) intersect in precisely \({\color{SeaGreen}p-1}\) elements.

Then the number of sets in \(\mathcal{F}\) is at most:

\({\color{IndianRed}|\mathcal{F}| \leqslant \left(\begin{array}{c} n \\ 0 \end{array}\right)+\left(\begin{array}{c} n \\ 1 \end{array}\right)+\cdots+\left(\begin{array}{c} n \\ p-1 \end{array}\right)}\)

17. Medium-Size Intersection Is Hard To Avoid

Let \(p\) be a prime number and let \(\mathcal{F}\) be a system
of \((2p-1)\) element subsets of an \(n\)-element set \(X\) such that
no two sets in \({\color{SeaGreen}\mathcal{F}}\) intersect in precisely \({\color{SeaGreen}p-1}\) elements.

A vector \(\mathbf{c}_A \in\{0,1\}^n\). This is simply the characteristic vector of \(A\), whose \(i\) th component is 1 if \(i \in A\) and 0 otherwise.

A function \(f_A:\{0,1\}^n \rightarrow \mathbb{F}_p\), given by
\(f_A(\mathbf{x}):=\prod_{s=0}^{p-2}\left(\left(\sum_{i \in A} x_i\right)-s\right)\).

With each set \(A \in \mathcal{F}\), associate:

All the arithmetic operations in the definition of \(f_A\) are in the finite field \(\mathbb{F}_p\), i.e., modulo \(p\) (and thus 0 and 1 are also treated as elements of \(\mathbb{F}_p\)).

17. Medium-Size Intersection Is Hard To Avoid

A vector \(\mathbf{c}_A \in\{0,1\}^n\). This is simply the characteristic vector of \(A\), whose \(i\) th component is 1 if \(i \in A\) and 0 otherwise.

A function \(f_A:\{0,1\}^n \rightarrow \mathbb{F}_p\), given by
\(f_A(\mathbf{x}):=\prod_{s=0}^{p-2}\left(\left(\sum_{i \in A} x_i\right)-s\right)\).

With each set \(A \in \mathcal{F}\), associate:

For example, for \(p=3, n=5\), and \(A=\{2,3\}\)

we have \(\mathbf{c}_A=(0,1,1,0,0)\) and \(f_A(\mathbf{x})=\left(x_2+x_3\right)\left(x_2+x_3-1\right)\).

17. Medium-Size Intersection Is Hard To Avoid

A vector \(\mathbf{c}_A \in\{0,1\}^n\). This is simply the characteristic vector of \(A\), whose \(i\) th component is 1 if \(i \in A\) and 0 otherwise.

A function \(f_A:\{0,1\}^n \rightarrow \mathbb{F}_p\), given by
\(f_A(\mathbf{x}):=\prod_{s=0}^{p-2}\left(\left(\sum_{i \in A} x_i\right)-s\right)\).

With each set \(A \in \mathcal{F}\), associate:

For example, for \(p=3, n=5\), \(A=\{2,3\}\), \(B = \{1,2,5\}\)
we have \(\mathbf{c}_A=(0,1,1,0,0)\), \(\mathbf{c}_B = (1,1,0,0,1)\),
\(f_A(\mathbf{x})=\left({\color{IndianRed}x_2+x_3}\right)\left(x_2+x_3-1\right)\) and \(f_A(\mathbf{c}_B)={\color{white}\left({\color{white}1+0}\right)\left({\color{white}1+0}-1\right).}\)

17. Medium-Size Intersection Is Hard To Avoid

A vector \(\mathbf{c}_A \in\{0,1\}^n\). This is simply the characteristic vector of \(A\), whose \(i\) th component is 1 if \(i \in A\) and 0 otherwise.

A function \(f_A:\{0,1\}^n \rightarrow \mathbb{F}_p\), given by
\(f_A(\mathbf{x}):=\prod_{s=0}^{p-2}\left(\left(\sum_{i \in A} x_i\right)-s\right)\).

With each set \(A \in \mathcal{F}\), associate:

For example, for \(p=3, n=5\), \(A=\{2,3\}\), \(B = \{1,2,5\}\)
we have \(\mathbf{c}_A=(0,1,1,0,0)\), \(\mathbf{c}_B = (1,1,0,0,1)\),
\(f_A(\mathbf{x})=\left({\color{IndianRed}x_2+x_3}\right)\left(x_2+x_3-1\right)\) and \(f_A(\mathbf{c}_B)=\left({\color{IndianRed}1+0}\right)\left({\color{IndianRed}1+0}-1\right).\)

17. Medium-Size Intersection Is Hard To Avoid

A vector \(\mathbf{c}_A \in\{0,1\}^n\). This is simply the characteristic vector of \(A\), whose \(i\) th component is 1 if \(i \in A\) and 0 otherwise.

A function \(f_A:\{0,1\}^n \rightarrow \mathbb{F}_p\), given by
\(f_A(\mathbf{x}):=\prod_{s=0}^{p-2}\left(\left(\sum_{i \in A} x_i\right)-s\right)\).

With each set \(A \in \mathcal{F}\), associate:

\(f_A(\mathbf{c}_B) = \prod_{s=0}^{p-2}(|A \cap B|-s)~~(\bmod p)\)

17. Medium-Size Intersection Is Hard To Avoid

We consider the set of all functions from \(\{0,1\}^d\) to \(\mathbb{F}_p\)
as a vector space over \(\mathbb{F}_p\) in the usual way,
and we let \(V_{\mathcal{F}}\) be the subspace spanned in it by the functions \(f_A, A \in \mathcal{F}\).

Part 1. We show that the \(f_A\)'s are linearly independent, and hence \({\color{IndianRed}\operatorname{dim}\left(V_{\mathcal{F}}\right)=|\mathcal{F}|}\).

Part 2. We will bound \({\color{SeaGreen}\operatorname{dim}(V_{\mathcal F})}\) from above.

17. Medium-Size Intersection Is Hard To Avoid

Part 1. We show that the \(f_A\)'s are linearly independent, and hence \({\color{IndianRed}\operatorname{dim}\left(V_{\mathcal{F}}\right)=|\mathcal{F}|}\).

\(f_A(\mathbf{c}_B)\begin{cases} \neq 0 & \text{if } |A \cap B| \equiv p-1 \bmod p,\\ = 0 & \text{if } |A \cap B| \not\equiv p-1 \bmod p.\end{cases}\)

\(f_A(\mathbf{c}_B) = \prod_{s=0}^{p-2}(|A \cap B|-s)~~(\bmod p)\)

17. Medium-Size Intersection Is Hard To Avoid

Part 1. We show that the \(f_A\)'s are linearly independent, and hence \({\color{IndianRed}\operatorname{dim}\left(V_{\mathcal{F}}\right)=|\mathcal{F}|}\).

\(f_A(\mathbf{c}_B)\begin{cases} = 0 & \text{if } 0 \leqslant |A \cap B| \leqslant p-2,\\\neq 0 & \text{if } |A \cap B| = p-1 \text{ or } |A \cap B| = 2p-1,\\ = 0 & \text{if } p \leqslant |A \cap B| \leqslant 2p-2.\end{cases}\)

\(f_A(\mathbf{c}_B) = \prod_{s=0}^{p-2}(|A \cap B|-s)~~(\bmod p)\)

17. Medium-Size Intersection Is Hard To Avoid

Part 1. We show that the \(f_A\)'s are linearly independent, and hence \({\color{IndianRed}\operatorname{dim}\left(V_{\mathcal{F}}\right)=|\mathcal{F}|}\).

\(f_A(\mathbf{c}_B) = \prod_{s=0}^{p-2}(|A \cap B|-s)~~(\bmod p)\)

For \(A \neq B\): \(0 \leqslant |A \cap B| \leqslant 2p-2\), and \(|A \cap B| \neq p-1\).

For \(A = B\): \(|A \cap B| = 2p - 1\).

17. Medium-Size Intersection Is Hard To Avoid

Part 1. We show that the \(f_A\)'s are linearly independent, and hence \({\color{IndianRed}\operatorname{dim}\left(V_{\mathcal{F}}\right)=|\mathcal{F}|}\).

\(f_A(\mathbf{c}_B) = \prod_{s=0}^{p-2}(|A \cap B|-s)~~(\bmod p)\)

\(\sum_{A \in \mathcal{F}} \alpha_A f_A=0\)

17. Medium-Size Intersection Is Hard To Avoid

Part 1. We show that the \(f_A\)'s are linearly independent, and hence \({\color{IndianRed}\operatorname{dim}\left(V_{\mathcal{F}}\right)=|\mathcal{F}|}\).

\(f_A(\mathbf{c}_B) = \prod_{s=0}^{p-2}(|A \cap B|-s)~~(\bmod p)\)

\(\sum_{A \in \mathcal{F}} \alpha_A f_A(\mathbf{c}_B)=0 \implies \alpha_B = 0\)

17. Medium-Size Intersection Is Hard To Avoid

Part 2. We will bound \({\color{SeaGreen}\operatorname{dim}(V_{\mathcal F})}\) from above.

In general, each \(f_A\) is a polynomial in \(x_1, x_2, \ldots, x_n\) of degree at most \(p-1\), and hence it is a linear combination of monomials of the form \(x_1^{i_1} x_2^{i_2} \cdots x_n^{i_n}, i_1+i_2+\cdots+i_n \leqslant p-1\).

We can still get rid of the monomials with some exponent \(i_j\) larger than 1 , because \(x_j^2\) and \(x_j\) represent the same function \(\{0,1\}^n \rightarrow \mathbb{F}_p\) (we substitute only 0's and 1's for the variables).

17. Medium-Size Intersection Is Hard To Avoid

Part 2. We will bound \({\color{SeaGreen}\operatorname{dim}(V_{\mathcal F})}\) from above.

where \(i_j \in \{0,1\}\) for all \(1 \leqslant j \leqslant n\).

\(\operatorname{dim}\left(V_{\mathcal{F}}\right) \leq\left(\begin{array}{l}n \\ 0\end{array}\right)+\left(\begin{array}{l}n \\ 1\end{array}\right)+\cdots+\left(\begin{array}{c}n \\ p-1\end{array}\right)\)

LinAlg 01

By Neeldhara Misra

LinAlg 01

Results about oddtown, eventown, same-sized, and medium-sized intersections, using dimensionality arguments.

LinAlg 01

More from Neeldhara Misra